SUMMARY: Understanding the mechanisms by which the living brain derives perception, cognition and behavior requires the ability to record and control electrical activity in many neurons simultaneously. Completely reconstructing the pattern of neural activity that mediates a specific neural operation is critical for fully understanding its underlying mechanism. This requires an approach that can measure neural activity on a large scale with high spatial and temporal resolution. Functional imaging of genetically encoded activity sensors is one of the most promising avenues towards achieving this goal because it permits dense sampling and unambiguous separation of nearby neurons1, 2. However, even techniques currently available cannot capture neural activity in large volumes of the brain simultaneously, at high speed, and with cellular resolution. Furthermore, these existing approaches rely on expensive and sophisticated hardware, and cannot be readily adapted for imaging in unrestrained animals. Since a central goal of the BRAIN initiative is to achieve large-scale recording of neural activity in behaving animals, a new imaging approach is needed to overcome these technical challenges. To achieve high speed, volumetric imaging in a compact and inexpensive device, we propose to develop a new imaging modality - compressive four-dimensional (4D) light field microscopy (LFM). In this approach, we will combine the advantages of light field microscopy with compressed sensing to extract the activity of thousands of individual neurons with high spatial and temporal resolution, through scattering tissue. In conventional microscopy, the photo-detector only samples the intensity of photons. In LFM, the sensor also captures the angle of the light. This allows 3D reconstruction, since position and angle enable back-tracing of rays of light. Such a scheme can be achieved simply by placing a lenslet array in the imaging pathway (Fig. 1A). The resulting 4D light field gives complete volumetric data at each time frame3-6. 3D activity can thus be sampled at camera-limited frame rates, much faster than conventional methods such as multiphoton or light sheet microscopy. For these reasons, LFM for functional brain imaging could revolutionize experimental neuroscience. Unfortunately, imaging methods which operate in the one-photon regime suffer from light scattering. To image activity in the mammalian brain, we must consider the effects of tissue scattering through the 3D volume. We propose to apply a new approach to processing light field data for better reconstructions of neurons through scattering media37,38. Since the effect of scattering is to spread the angles of propagation of light, measuring angle information inherently helps to characterize and mitigate scatter effects. Our method leverages compressed sensing algorithms, which exploit the sparsity of the light emission in 3D space and time. Importantly, the volume and resolution limits of our method are not set by the number of pixels captured, but rather by the number of active neurons at any given time. Thus, we will be able to localize and measure the activity of thousands or millions of individually active neurons in large volumes of cerebral cortical tissue.